From 3f18e92896d390ea50c71ce2b9e84ad242e1ddc2 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 3 Mar 2026 17:00:01 +0100 Subject: [PATCH] [PS] Probably summarized all of tomorrow's content --- .../01_random-variables/04_construction.tex | 33 +++++++++++++++--- .../probability-and-statistics-cheatsheet.pdf | Bin 294966 -> 300921 bytes 2 files changed, 29 insertions(+), 4 deletions(-) diff --git a/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex b/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex index 3ceef16..13a4576 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex @@ -1,6 +1,31 @@ -% P28 +% P28: TODO: may need expansion \subsection{Konstruktion von Zufallsvariablen} -\shortdefinition[Bernoulli ZV] mit param. $p \in [0, 1]$ falls $\P[\cX = 0] = 1 - p$ und $\P[\cX = 1] = p$. -Wir schreiben: $\cX \sim \text{Ber}(p)$ +\shortdefinition[Bernoulli ZV] mit param. $p \in [0, 1]$ falls\\ +$\P[\cX = 0] = 1 - p$ und $\P[\cX = 1] = p$. Not.: $\cX \sim \text{Ber}(p)$ -\shorttheorem[$\exists$-T v. Kolmogorov] +\shorttheorem[$\exists$-T v. Kolmogorov] $\exists$ W-Raum und n. endl. uiv Folge von $\cX_i \sim \text{Ber}\left( \frac{1}{2} \right)$ + +\shortdefinition ZV $U$ heisst \bi{gleichverteilt auf} [0, 1], falls\\ +$F_U(x) = \begin{cases} + 0 & x < 0 \\ + x & 0 \leq x \leq 1 \\ + 1 & x > 1 + \end{cases}$. +Wir schreiben $U \sim \cU([0, 1])$ + +\shorttheorem $\cX_i$ wie oben, da $\cX_i(\omega) \in \{ 0, 1 \}$ konvergiert +$\cY(\omega) = \sum_{n = 1}^{\8} 2^{-n} \cX_n (\omega)$ absolut, mit $\cY(\omega) \in [0, 1]$. +$\cY \sim \cU([0, 1])$ + +\shortdefinition[Pseudoinverse] von $F$ ist $F^{-1} : (0, 1) \rightarrow \R$. Def:\\ +$\forall \alpha \in (0, 1) \quad F^{-1} (\alpha) = \inf\{ x \in \R \divider F(x) \geq \alpha \}$ + +\shorttheorem[Inversionsmethode] $F$ erfüllt eig. v. T2.4, $U \sim \cU(\dots)$, dann hat ZV $X = F^{-1}(U)$ die Verteilfunk. $F_X = F$ + +\shortremark $X$ wohldefiniert mit $X(\omega) = F^{-1}(U(\omega))$ falls $U(\omega) \in (0, 1)$ und $X(\omega) = 0$ sonst. + +\shorttheorem $F_1, \ldots$ Folge von Funk. mit eig. v. T2.4. Dann $\exists$ W-Raum und Folge von unabhängigen ZV $X_i$, sodass: +\begin{itemize} + \item $\forall i \; X_i$ has $F_i$ (also $\forall x\; \P[X_i \leq x] = F_i(x)$) + \item $X_1, X_2, \ldots$ sind unabhängig. +\end{itemize} diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 8b8c70da816f768688bc51244a0b9dbc74b796ad..1e2393902c9bce11b1022e3cc68a444083a86536 100644 GIT binary patch delta 144076 zcmV(^K-IstfD-wx60pNC12r-^m!M(%5FACW8A#yV*KP*-m*= zm6JHK!$(CaDZWB$dP1k600f>DGUZ^G=Q%!Xf)~QWJE{b z?tHVq^W`%}?Z^?QxgPBwkF>FbGd9wc5^2#^ z)ywan_%E{hX@8ehPxqgH`!xRj>#N;&`~Tc|y1(D0HNpCfd5u;l<9I zcXV_F&;B{0L^wP8J=DAyS!#$i62700e%kqQht|Y4pKEQ7nIO~*YBLPn(q!C{St6Aj zjkzLR$T)5u#?d99`TX*kR3j#dkbj!{CdQI7BBhRlWEN{<^k&R}mDR5|$CLB(#rw&w zVAVCSw<3>$d0JPmb_K1b*URbb{APZ#IKG*mEoYbW={##E_RWY>D)dc(3zjg!>rM$n zyZ!oLvAmv~EvLROQ$3l!rA*FezJq4Ux)UY@5yE)p{%Zv7eLD(`?gaiSV1KYL_D_f& 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